Let $G_K$ be an absolute Galois group of algebraic field $K$. Let $M$ be an finite abelian group. Let $G$ acts continuously on $M$($G$ with krull topology and $M$ with discrete one), then I want to prove there exists some finite index normal subgroup of $G_K$ which fixes all elements of $M$.
I know for all $x∈M$, $stab(x)$ is finite subgroup of $G_K$(I assume this fact here), so from here, If I could find normal subgroup which contains union of $stab(x),x∈M$, that is the normal subgroup which I wanted.
But I don't have confident that I can take such normal subgroup. Another approach are also welcomed, thank you for your help.